Let $xinBbb R$, $a=inf {r mid rinBbb Q,x<r}$, and $b=inf{-smid sinBbb Q,s<x}$. Prove that $a+b=inf{r-s...
Let $x,yinBbb R$. We define addition operation $(+)$ by $$x+ y:=inf{r+ smid r,sinBbb Q text{ and } x<r text{ and } y<s}$$
Theorem: Let $xinBbb R$, $a=inf {r mid rinBbb Q,x<r}$, and $b=inf{-smid sinBbb Q,s<x}$. Prove that $$a+b=inf{r-s mid r,sinBbb Q,s<x<r}$$
This is a theorem that the authors of my textbook Introduction to Set Theory by Hrbacek and Jech leave as an exercise. The theorem is from Chapter 10. Sets of Real Numbers. Could you please verify my attempt? Thank you for your help!
My attempt:
It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$.
By definition, $a+b=inf{r+pmid r,pinBbb Q,a<r,b<p}=inf{r+pmid r,pinBbb Q,x<r,b<p}$.
Substituting $-p$ for $p$, we get $a+b=inf{r-pmid r,-pinBbb Q,x<r,b<-p}=$ $inf{r-pmid r,pinBbb Q,x<r,b<-p}$.
We have $pinBbb Q$ and $b<-p iff pinBbb Q$ and $-p>-s$ for some $sinBbb Q$ such that $s<x$ $iff pinBbb Q$ and $p<s$ for some $sinBbb Q$ such that $s<x$ $iff p=s$ for some $sinBbb Q$ such that $s<x$.
It follows that $a+b=inf{r-pmid r,pinBbb Q,x<r,b<-p}=$ $inf{r-smid r,sinBbb Q,x<r,s<x}=inf{r-smid r,sinBbb Q,s<x<r}.$
proof-verification real-numbers
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
|
show 2 more comments
Let $x,yinBbb R$. We define addition operation $(+)$ by $$x+ y:=inf{r+ smid r,sinBbb Q text{ and } x<r text{ and } y<s}$$
Theorem: Let $xinBbb R$, $a=inf {r mid rinBbb Q,x<r}$, and $b=inf{-smid sinBbb Q,s<x}$. Prove that $$a+b=inf{r-s mid r,sinBbb Q,s<x<r}$$
This is a theorem that the authors of my textbook Introduction to Set Theory by Hrbacek and Jech leave as an exercise. The theorem is from Chapter 10. Sets of Real Numbers. Could you please verify my attempt? Thank you for your help!
My attempt:
It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$.
By definition, $a+b=inf{r+pmid r,pinBbb Q,a<r,b<p}=inf{r+pmid r,pinBbb Q,x<r,b<p}$.
Substituting $-p$ for $p$, we get $a+b=inf{r-pmid r,-pinBbb Q,x<r,b<-p}=$ $inf{r-pmid r,pinBbb Q,x<r,b<-p}$.
We have $pinBbb Q$ and $b<-p iff pinBbb Q$ and $-p>-s$ for some $sinBbb Q$ such that $s<x$ $iff pinBbb Q$ and $p<s$ for some $sinBbb Q$ such that $s<x$ $iff p=s$ for some $sinBbb Q$ such that $s<x$.
It follows that $a+b=inf{r-pmid r,pinBbb Q,x<r,b<-p}=$ $inf{r-smid r,sinBbb Q,x<r,s<x}=inf{r-smid r,sinBbb Q,s<x<r}.$
proof-verification real-numbers
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
2
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
1
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20
|
show 2 more comments
Let $x,yinBbb R$. We define addition operation $(+)$ by $$x+ y:=inf{r+ smid r,sinBbb Q text{ and } x<r text{ and } y<s}$$
Theorem: Let $xinBbb R$, $a=inf {r mid rinBbb Q,x<r}$, and $b=inf{-smid sinBbb Q,s<x}$. Prove that $$a+b=inf{r-s mid r,sinBbb Q,s<x<r}$$
This is a theorem that the authors of my textbook Introduction to Set Theory by Hrbacek and Jech leave as an exercise. The theorem is from Chapter 10. Sets of Real Numbers. Could you please verify my attempt? Thank you for your help!
My attempt:
It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$.
By definition, $a+b=inf{r+pmid r,pinBbb Q,a<r,b<p}=inf{r+pmid r,pinBbb Q,x<r,b<p}$.
Substituting $-p$ for $p$, we get $a+b=inf{r-pmid r,-pinBbb Q,x<r,b<-p}=$ $inf{r-pmid r,pinBbb Q,x<r,b<-p}$.
We have $pinBbb Q$ and $b<-p iff pinBbb Q$ and $-p>-s$ for some $sinBbb Q$ such that $s<x$ $iff pinBbb Q$ and $p<s$ for some $sinBbb Q$ such that $s<x$ $iff p=s$ for some $sinBbb Q$ such that $s<x$.
It follows that $a+b=inf{r-pmid r,pinBbb Q,x<r,b<-p}=$ $inf{r-smid r,sinBbb Q,x<r,s<x}=inf{r-smid r,sinBbb Q,s<x<r}.$
proof-verification real-numbers
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
Let $x,yinBbb R$. We define addition operation $(+)$ by $$x+ y:=inf{r+ smid r,sinBbb Q text{ and } x<r text{ and } y<s}$$
Theorem: Let $xinBbb R$, $a=inf {r mid rinBbb Q,x<r}$, and $b=inf{-smid sinBbb Q,s<x}$. Prove that $$a+b=inf{r-s mid r,sinBbb Q,s<x<r}$$
This is a theorem that the authors of my textbook Introduction to Set Theory by Hrbacek and Jech leave as an exercise. The theorem is from Chapter 10. Sets of Real Numbers. Could you please verify my attempt? Thank you for your help!
My attempt:
It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$.
By definition, $a+b=inf{r+pmid r,pinBbb Q,a<r,b<p}=inf{r+pmid r,pinBbb Q,x<r,b<p}$.
Substituting $-p$ for $p$, we get $a+b=inf{r-pmid r,-pinBbb Q,x<r,b<-p}=$ $inf{r-pmid r,pinBbb Q,x<r,b<-p}$.
We have $pinBbb Q$ and $b<-p iff pinBbb Q$ and $-p>-s$ for some $sinBbb Q$ such that $s<x$ $iff pinBbb Q$ and $p<s$ for some $sinBbb Q$ such that $s<x$ $iff p=s$ for some $sinBbb Q$ such that $s<x$.
It follows that $a+b=inf{r-pmid r,pinBbb Q,x<r,b<-p}=$ $inf{r-smid r,sinBbb Q,x<r,s<x}=inf{r-smid r,sinBbb Q,s<x<r}.$
proof-verification real-numbers
proof-verification real-numbers
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
edited Jan 4 at 6:35
Le Anh Dung
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
asked Jan 4 at 5:30
Le Anh DungLe Anh Dung
1,0301521
1,0301521
Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
2
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
1
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20
|
show 2 more comments
Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
2
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
1
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20
Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
2
2
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
1
1
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20
|
show 2 more comments
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Hi @KaviRamaMurthy, I could not see the mentioned typos. Could you please be more specific?
– Le Anh Dung
Jan 4 at 6:30
Which book are you referring to? Please edit the question to include the details.
– Shaun
Jan 4 at 6:32
2
Thank you @Shaun! I have edited to include the name of textbook.
– Le Anh Dung
Jan 4 at 6:36
This exercise is about construction of the real number. When you say It is clear that $a=x$ from $a=inf {r mid rinBbb Q,x<r}$, do you mean it is clear from your knowledge of the reals (which won’t be fine!) or clear from theorems proved previously in the exercise (which would be OK). I doubt that you’re in the later case as you won’t have proved existence and unicity of infimum before defining addition. Which is the exercise about. For this exercise, forget all what you know about the real numbers!
– mathcounterexamples.net
Jan 4 at 7:10
1
@LeAnhDung OK, understand! And what you did seems fine.
– mathcounterexamples.net
Jan 4 at 8:20